For Fano reductive group compactifications, the Donaldson-Futaki invariant of an affine-linear test configuration equals half the inner product of the function's gradient with the difference between the polytope's barycenter and 2ρ.
Ruzzi, Fano symmetric varieties with low rank , Publ
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A technical remark on the Donaldson-Futaki invariant for Fano reductive group compactifications
For Fano reductive group compactifications, the Donaldson-Futaki invariant of an affine-linear test configuration equals half the inner product of the function's gradient with the difference between the polytope's barycenter and 2ρ.